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			<h1 id="firstHeading" class="firstHeading">
				<span dir="auto">Step response</span>
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<div class="thumbinner" style="width:302px;"><a href="http://en.wikipedia.org/wiki/File:High_accuracy_settling_time_measurements_figure_1.png" class="image"><img alt="" src="wikipedia-Step_response_pliki/300px-High_accuracy_settling_time_measurements_figure_1.png" class="thumbimage" width="300" height="219"></a>
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<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:High_accuracy_settling_time_measurements_figure_1.png" class="internal" title="Enlarge"><img src="wikipedia-Step_response_pliki/magnify-clip.png" alt="" width="15" height="11"></a></div>
A typical step response for a second order system, illustrating <a href="http://en.wikipedia.org/wiki/Overshoot_%28signal%29" title="Overshoot (signal)">overshoot</a>, followed by <a href="http://en.wikipedia.org/wiki/Ringing_%28signal%29" title="Ringing (signal)">ringing</a>, all subsiding within a <a href="http://en.wikipedia.org/wiki/Settling_time" title="Settling time">settling time</a>.</div>
</div>
</div>
<p>The <b>step response</b> of a system in a given initial state consists of the time evolution of its outputs when its control inputs are <a href="http://en.wikipedia.org/wiki/Heaviside_step_function" title="Heaviside step function">Heaviside step functions</a>. In <a href="http://en.wikipedia.org/wiki/Electronic_engineering" title="Electronic engineering">electronic engineering</a> and <a href="http://en.wikipedia.org/wiki/Control_theory" title="Control theory">control theory</a>, step response is the time behaviour of the outputs of a general <a href="http://en.wikipedia.org/wiki/System" title="System">system</a>
 when its inputs change from zero to one in a very short time. The 
concept can be extended to the abstract mathematical notion of a <a href="http://en.wikipedia.org/wiki/Dynamical_system" title="Dynamical system">dynamical system</a> using an <a href="http://en.wikipedia.org/wiki/Dynamical_system_%28definition%29#General_definition" title="Dynamical system (definition)">evolution parameter</a>.</p>
<p>From a practical standpoint, knowing how the system responds to a 
sudden input is important because large and possibly fast deviations 
from the long term steady state may have extreme effects on the 
component itself and on other portions of the overall system dependent 
on this component. In addition, the overall system cannot act until the 
component's output settles down to some vicinity of its final state, 
delaying the overall system response. Formally, knowing the step 
response of a dynamical system gives information on the <a href="http://en.wikipedia.org/wiki/Stability_theory" title="Stability theory">stability</a> of such a system, and on its ability to reach one stationary state when starting from another.</p>
<table id="toc" class="toc">
<tbody><tr>
<td>
<div id="toctitle">
<h2>Contents</h2>
<span class="toctoggle">&nbsp;[<a href="#" class="internal" id="togglelink">hide</a>]&nbsp;</span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#Time_domain_versus_frequency_domain"><span class="tocnumber">1</span> <span class="toctext">Time domain <i>versus</i> frequency domain</span></a></li>
<li class="toclevel-1 tocsection-2"><a href="#Step_response_of_feedback_amplifiers"><span class="tocnumber">2</span> <span class="toctext">Step response of feedback amplifiers</span></a>
<ul>
<li class="toclevel-2 tocsection-3"><a href="#Analysis"><span class="tocnumber">2.1</span> <span class="toctext">Analysis</span></a></li>
<li class="toclevel-2 tocsection-4"><a href="#Results"><span class="tocnumber">2.2</span> <span class="toctext">Results</span></a></li>
<li class="toclevel-2 tocsection-5"><a href="#Control_of_overshoot"><span class="tocnumber">2.3</span> <span class="toctext">Control of overshoot</span></a></li>
<li class="toclevel-2 tocsection-6"><a href="#Control_of_settling_time"><span class="tocnumber">2.4</span> <span class="toctext">Control of settling time</span></a></li>
<li class="toclevel-2 tocsection-7"><a href="#Phase_margin"><span class="tocnumber">2.5</span> <span class="toctext">Phase margin</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-8"><a href="#Formal_mathematical_description"><span class="tocnumber">3</span> <span class="toctext">Formal mathematical description</span></a>
<ul>
<li class="toclevel-2 tocsection-9"><a href="#Nonlinear_dynamical_system"><span class="tocnumber">3.1</span> <span class="toctext">Nonlinear dynamical system</span></a></li>
<li class="toclevel-2 tocsection-10"><a href="#Linear_dynamical_system"><span class="tocnumber">3.2</span> <span class="toctext">Linear dynamical system</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-11"><a href="#See_also"><span class="tocnumber">4</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-12"><a href="#References_and_notes"><span class="tocnumber">5</span> <span class="toctext">References and notes</span></a></li>
<li class="toclevel-1 tocsection-13"><a href="#Further_reading"><span class="tocnumber">6</span> <span class="toctext">Further reading</span></a></li>
<li class="toclevel-1 tocsection-14"><a href="#External_links"><span class="tocnumber">7</span> <span class="toctext">External links</span></a></li>
</ul>
</td>
</tr>
</tbody></table>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Step_response&amp;action=edit&amp;section=1" title="Edit section: Time domain versus frequency domain">edit</a>]</span> <span class="mw-headline" id="Time_domain_versus_frequency_domain">Time domain <i>versus</i> frequency domain</span></h2>
<p>Instead of frequency response, system performance may be specified in
 terms of parameters describing time-dependence of response. The step 
response can be described by the following quantities related to its <b>time behavior</b>,</p>
<ul>
<li><a href="http://en.wikipedia.org/wiki/Overshoot_%28signal%29" title="Overshoot (signal)">overshoot</a></li>
<li><a href="http://en.wikipedia.org/wiki/Rise_time" title="Rise time">rise time</a></li>
<li><a href="http://en.wikipedia.org/wiki/Settling_time" title="Settling time">settling time</a></li>
<li><a href="http://en.wikipedia.org/wiki/Ringing_%28signal%29" title="Ringing (signal)">ringing</a></li>
</ul>
<p>In the case of <a href="http://en.wikipedia.org/wiki/Linear" title="Linear" class="mw-redirect">linear</a> dynamic systems, much can be inferred about the system from these characteristics. <a href="http://en.wikipedia.org/wiki/Step_response#Results" title="Step response">Below</a> the step response of a simple two-pole amplifier is presented, and some of these terms are illustrated.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Step_response&amp;action=edit&amp;section=2" title="Edit section: Step response of feedback amplifiers">edit</a>]</span> <span class="mw-headline" id="Step_response_of_feedback_amplifiers">Step response of feedback amplifiers</span></h2>
<div class="thumb tright">
<div class="thumbinner" style="width:302px;"><a href="http://en.wikipedia.org/wiki/File:Block_Diagram_for_Feedback.svg" class="image"><img alt="" src="wikipedia-Step_response_pliki/300px-Block_Diagram_for_Feedback.png" class="thumbimage" width="300" height="139"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Block_Diagram_for_Feedback.svg" class="internal" title="Enlarge"><img src="wikipedia-Step_response_pliki/magnify-clip.png" alt="" width="15" height="11"></a></div>
Figure 1: Ideal negative feedback model; open loop gain is <i>A</i><sub>OL</sub> and feedback factor is β.</div>
</div>
</div>
<p>This section describes the step response of a simple <a href="http://en.wikipedia.org/wiki/Negative_feedback_amplifier" title="Negative feedback amplifier">negative feedback amplifier</a> shown in Figure 1. The feedback amplifier consists of a main <b>open-loop</b> amplifier of gain <i>A</i><sub>OL</sub> and a feedback loop governed by a <b>feedback factor</b>
 β. This feedback amplifier is analyzed to determine how its step 
response depends upon the time constants governing the response of the 
main amplifier, and upon the amount of feedback used.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Step_response&amp;action=edit&amp;section=3" title="Edit section: Analysis">edit</a>]</span> <span class="mw-headline" id="Analysis">Analysis</span></h3>
<p>A negative feedback amplifier has gain given by (see <a href="http://en.wikipedia.org/wiki/Negative_feedback_amplifier" title="Negative feedback amplifier">negative feedback amplifier</a>):</p>
<dl>
<dd><img class="tex" alt="A_{FB} = \frac {A_{OL}} {1+ \beta A_{OL}}, " src="wikipedia-Step_response_pliki/eb18ac23679b2d13b793a7d6defd4b14.png"></dd>
</dl>
<p>where <i>A</i><sub>OL</sub> = <b>open-loop</b> gain, <i>A</i><sub>FB</sub> = <b>closed-loop</b> gain (the gain with negative feedback present) and β = <b>feedback factor</b>. The step response of such an amplifier is easily handled in the case that the open-loop gain has two poles (two <a href="http://en.wikipedia.org/wiki/Time_constant" title="Time constant">time constants</a>, τ<sub>1</sub>, τ<sub>2</sub>), that is, the open-loop gain is given by:</p>
<dl>
<dd><img class="tex" alt="A_{OL} = \frac {A_0} {(1+j \omega \tau_1) (1 + j \omega \tau_2)}, " src="wikipedia-Step_response_pliki/918223cdc541dd21f5cdf0f681dd86d6.png"></dd>
</dl>
<p>with zero-frequency gain <i>A</i><sub>0</sub> and angular frequency ω = 2π<i>f</i>, which leads to the closed-loop gain:</p>
<dl>
<dd><img class="tex" alt="A_{FB} = \frac {A_0} {1+ \beta A_0}" src="wikipedia-Step_response_pliki/c8bc25fc0ea87e1341600fae1fb7bc85.png">   •   <img class="tex" alt=" \  \frac {1} {1+j \omega \frac { \tau_1 + \tau_2 } {1 + \beta A_0} + (j \omega )^2 \frac { \tau_1 \tau_2} {1 + \beta A_0} }.  " src="wikipedia-Step_response_pliki/efffe34f44e818a9ae526142447bc455.png"></dd>
</dl>
<div class="thumb tright">
<div class="thumbinner" style="width:252px;"><a href="http://en.wikipedia.org/wiki/File:Conjugate_poles_in_s-plane.PNG" class="image"><img alt="" src="wikipedia-Step_response_pliki/250px-Conjugate_poles_in_s-plane.PNG" class="thumbimage" width="250" height="237"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Conjugate_poles_in_s-plane.PNG" class="internal" title="Enlarge"><img src="wikipedia-Step_response_pliki/magnify-clip.png" alt="" width="15" height="11"></a></div>
Figure 2: Conjugate pole locations for a two-pole feedback amplifier; <i>Re</i>(s) = real axis and <i>Im</i>(s) = imaginary axis.</div>
</div>
</div>
<p>The time dependence of the amplifier is easy to discover by switching variables to <i>s</i> = <i>j</i>ω, whereupon the gain becomes:</p>
<dl>
<dd><img class="tex" alt=" A_{FB} = \frac {A_0} { \tau_1 \tau_2 }" src="wikipedia-Step_response_pliki/789130a046b134599afbc5567d1cf744.png">   •   <img class="tex" alt=" \frac {1} {s^2 +s \left( \frac {1} {\tau_1} + \frac {1} {\tau_2} \right) + \frac {1+ \beta A_0} {\tau_1 \tau_2}} " src="wikipedia-Step_response_pliki/03daa3f02194ac1513f150770ba07671.png"></dd>
</dl>
<p>The poles of this expression (that is, the zeros of the denominator) occur at:</p>
<dl>
<dd><img class="tex" alt="2s = - \left( \frac {1} {\tau_1} + \frac {1} {\tau_2} \right) " src="wikipedia-Step_response_pliki/d4de2e1b4778adc0a7b171dde65266d2.png">
<dl>
<dd>
<dl>
<dd>
<dl>
<dd><img class="tex" alt="\pm \sqrt { \left( \frac {1} {\tau_1} - \frac {1} {\tau_2} \right) ^2 -\frac {4 \beta A_0 } {\tau_1 \tau_2 } }," src="wikipedia-Step_response_pliki/c9406e37399d9f24778c76a148c16cfc.png"></dd>
</dl>
</dd>
</dl>
</dd>
</dl>
</dd>
</dl>
<p>which shows for large enough values of β<i>A</i><sub>0</sub> the 
square root becomes the square root of a negative number, that is the 
square root becomes imaginary, and the pole positions are complex 
conjugate numbers, either <i>s</i><sub>+</sub> or <i>s</i><sub>−</sub>; see Figure 2:</p>
<dl>
<dd><img class="tex" alt=" s_{\pm} = -\rho \pm j \mu, \,  " src="wikipedia-Step_response_pliki/ce00b703123899e70163fd5b118feb1a.png"></dd>
</dl>
<p>with</p>
<dl>
<dd>
<dl>
<dd><img class="tex" alt=" \rho = \frac {1}{2} \left( \frac {1} {\tau_1} + \frac {1} {\tau_2} \right ), " src="wikipedia-Step_response_pliki/8449b28f5229db6c129f0bbb919561e1.png"></dd>
</dl>
</dd>
</dl>
<p>and</p>
<dl>
<dd>
<dl>
<dd><img class="tex" alt=" \mu = \frac {1} {2} \sqrt { \frac {4 \beta A_0} { \tau_1 \tau_2} - \left( \frac {1} {\tau_1} - \frac {1} {\tau_2} \right)^2 }. " src="wikipedia-Step_response_pliki/b882c6bdbd8dc061f171115049f15741.png"></dd>
</dl>
</dd>
</dl>
<p>Using polar coordinates with the magnitude of the radius to the roots given by |<i>s</i>| (Figure 2):</p>
<dl>
<dd><img class="tex" alt=" | s | = |s_{ \pm } | =  \sqrt{ \rho^2 +\mu^2}, " src="wikipedia-Step_response_pliki/a22ba9cfacfce81690b19fc1664c43e7.png"></dd>
</dl>
<p>and the angular coordinate φ is given by:</p>
<dl>
<dd><img class="tex" alt=" \cos \phi = \frac { \rho} { | s | } , \sin \phi = \frac { \mu} { | s | }." src="wikipedia-Step_response_pliki/b4d551ab0ed1ce2dbcbf84746bbfa76d.png"></dd>
</dl>
<p>Tables of <a href="http://en.wikipedia.org/wiki/Laplace_transforms" title="Laplace transforms" class="mw-redirect">Laplace transforms</a> show that the time response of such a system is composed of combinations of the two functions:</p>
<dl>
<dd>
<dl>
<dd><img class="tex" alt=" e^{- \rho t} \sin ( \mu t) \quad\text{and} \quad " src="wikipedia-Step_response_pliki/2c05356e02764ce2f68875eadfced6e7.png"> <img class="tex" alt=" e^{- \rho t} \cos ( \mu t), " src="wikipedia-Step_response_pliki/43ba92ea6305b63472901a7cbe816e4f.png"></dd>
</dl>
</dd>
</dl>
<p>which is to say, the solutions are damped oscillations in time. In particular, the unit step response of the system is:<sup id="cite_ref-Kuo_0-0" class="reference"><a href="#cite_note-Kuo-0"><span>[</span>1<span>]</span></a></sup></p>
<dl>
<dd><img class="tex" alt="S(t) = 1 -   e^{- \rho t} \  \frac {  \sin \left( \mu t + \phi \right)}{ \sin( \phi )} \ . " src="wikipedia-Step_response_pliki/47d24e8b964c6dd6d979fbb892358da1.png"></dd>
</dl>
<p>Notice that the damping of the response is set by ρ, that is, by the 
time constants of the open-loop amplifier. In contrast, the frequency of
 oscillation is set by μ, that is, by the feedback parameter through β<i>A</i><sub>0</sub>. Because ρ is a sum of reciprocals of time constants, it is interesting to notice that ρ is dominated by the <i>shorter</i> of the two.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Step_response&amp;action=edit&amp;section=4" title="Edit section: Results">edit</a>]</span> <span class="mw-headline" id="Results">Results</span></h3>
<div class="thumb tright">
<div class="thumbinner" style="width:352px;"><a href="http://en.wikipedia.org/wiki/File:Step_response_for_two-pole_feedback_amplifier.PNG" class="image"><img alt="" src="wikipedia-Step_response_pliki/350px-Step_response_for_two-pole_feedback_amplifier.PNG" class="thumbimage" width="350" height="373"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Step_response_for_two-pole_feedback_amplifier.PNG" class="internal" title="Enlarge"><img src="wikipedia-Step_response_pliki/magnify-clip.png" alt="" width="15" height="11"></a></div>
Figure 3: Step-response of a linear two-pole feedback amplifier; time is
 in units of&nbsp;1/ρ, that is, in terms of the time constants of <i>A</i><sub>OL</sub>; curves are plotted for three values of <i>mu</i>&nbsp;=&nbsp;<i>μ</i>, which is controlled by&nbsp;β.</div>
</div>
</div>
<p>Figure 3 shows the time response to a unit step input for three 
values of the parameter μ. It can be seen that the frequency of 
oscillation increases with μ, but the oscillations are contained between
 the two asymptotes set by the exponentials 
[&nbsp;1&nbsp;−&nbsp;exp&nbsp;(−ρt)&nbsp;] and 
[&nbsp;1&nbsp;+&nbsp;exp(−ρt)&nbsp;]. These asymptotes are determined by
 ρ and therefore by the time constants of the open-loop amplifier, 
independent of feedback.</p>
<p>The phenomena of oscillation about final value is called <b><a href="http://en.wikipedia.org/wiki/Ringing_%28signal%29" title="Ringing (signal)">ringing</a></b>. The <b><a href="http://en.wikipedia.org/wiki/Overshoot_%28signal%29" title="Overshoot (signal)">overshoot</a></b> is the maximum swing above final value, and clearly increases with μ. Likewise, the <b>undershoot</b> is the minimum swing below final value, again increasing with μ. The <b><a href="http://en.wikipedia.org/wiki/Settling_time" title="Settling time">settling time</a></b> is the time for departures from final value to sink below some specified level, say 10% of final value.</p>
<p>The dependence of settling time upon μ is not obvious, and the 
approximation of a two-pole system probably is not accurate enough to 
make any real-world conclusions about feedback dependence of settling 
time. However, the asymptotes [&nbsp;1&nbsp;−&nbsp;exp&nbsp;(−ρt)&nbsp;]
 and [&nbsp;1&nbsp;+&nbsp;exp&nbsp;(−ρt)&nbsp;] clearly impact settling 
time, and they are controlled by the time constants of the open-loop 
amplifier, particularly the shorter of the two time constants. That 
suggests that a specification on settling time must be met by 
appropriate design of the open-loop amplifier.</p>
<p>The two major conclusions from this analysis are:</p>
<ol>
<li>Feedback controls the amplitude of oscillation about final value for
 a given open-loop amplifier and given values of open-loop time 
constants, τ<sub>1</sub> and τ<sub>2</sub>.</li>
<li>The open-loop amplifier decides settling time. It sets the time 
scale of Figure 3, and the faster the open-loop amplifier, the faster 
this time scale.</li>
</ol>
<p>As an aside, it may be noted that real-world departures from this 
linear two-pole model occur due to two major complications: first, real 
amplifiers have more than two poles, as well as zeros; and second, real 
amplifiers are nonlinear, so their step response changes with signal 
amplitude.</p>
<div class="thumb tright">
<div class="thumbinner" style="width:302px;"><a href="http://en.wikipedia.org/wiki/File:Overshoot_control.PNG" class="image"><img alt="" src="wikipedia-Step_response_pliki/300px-Overshoot_control.PNG" class="thumbimage" width="300" height="421"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Overshoot_control.PNG" class="internal" title="Enlarge"><img src="wikipedia-Step_response_pliki/magnify-clip.png" alt="" width="15" height="11"></a></div>
Figure 4: Step response for three values of&nbsp;α. Top: α 
&nbsp;=&nbsp;4; Center: α&nbsp;=&nbsp;2; Bottom: α&nbsp;=&nbsp;0.5. As α
 is reduced the pole separation reduces, and the overshoot increases.</div>
</div>
</div>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Step_response&amp;action=edit&amp;section=5" title="Edit section: Control of overshoot">edit</a>]</span> <span class="mw-headline" id="Control_of_overshoot">Control of overshoot</span></h3>
<p>How overshoot may be controlled by appropriate parameter choices is discussed next.</p>
<p>Using the equations above, the amount of overshoot can be found by 
differentiating the step response and finding its maximum value. The 
result for maximum step response <i>S</i><sub>max</sub> is:<sup id="cite_ref-Kuo2_1-0" class="reference"><a href="#cite_note-Kuo2-1"><span>[</span>2<span>]</span></a></sup></p>
<dl>
<dd><img class="tex" alt="S_\max= 1 + \exp \left( - \pi \frac { \rho }{ \mu } \right). " src="wikipedia-Step_response_pliki/f4b9611d3c31ab6d4ec01384c92ec1c8.png"></dd>
</dl>
<p>The final value of the step response is 1, so the exponential is the 
actual overshoot itself. It is clear the overshoot is zero if μ = 0, 
which is the condition:</p>
<dl>
<dd><img class="tex" alt=" \frac {4 \beta A_0} { \tau_1 \tau_2} = \left( \frac {1} {\tau_1} - \frac {1} {\tau_2} \right)^2. " src="wikipedia-Step_response_pliki/95fe094586c8254ac259c9faf322d76a.png"></dd>
</dl>
<p>This quadratic is solved for the ratio of time constants by setting <i>x</i> = ( τ<sub>1</sub> / τ<sub>2</sub> )<sup>1 / 2</sup> with the result</p>
<dl>
<dd><img class="tex" alt="x = \sqrt{ \beta A_0 } + \sqrt { \beta A_0 +1 }. \, " src="wikipedia-Step_response_pliki/22966983d296e7eda3be122fb7c35621.png"></dd>
</dl>
<p>Because β <i>A</i><sub>0</sub> &gt;&gt; 1, the 1 in the square root can be dropped, and the result is</p>
<dl>
<dd><img class="tex" alt=" \frac { \tau_1} { \tau_2} = 4 \beta A_0. " src="wikipedia-Step_response_pliki/cbb2f3623794604f2f95cde8a36a6ab5.png"></dd>
</dl>
<p>In words, the first time constant must be much larger than the 
second. To be more adventurous than a design allowing for no overshoot 
we can introduce a factor α in the above relation:</p>
<dl>
<dd><img class="tex" alt=" \frac { \tau_1} { \tau_2} = \alpha \beta A_0, " src="wikipedia-Step_response_pliki/b46bd9857594dce5f1310735ef8a22e9.png"></dd>
</dl>
<p>and let α be set by the amount of overshoot that is acceptable.</p>
<p>Figure 4 illustrates the procedure. Comparing the top panel (α = 4) 
with the lower panel (α&nbsp;=&nbsp;0.5) shows lower values for α 
increase the rate of response, but increase overshoot. The case 
α&nbsp;=&nbsp;2 (center panel) is the <a href="http://en.wikipedia.org/wiki/Butterworth_filter#Maximal_flatness" title="Butterworth filter"><i>maximally flat</i></a> design that shows no peaking in the <a href="http://en.wikipedia.org/wiki/Bode_plot" title="Bode plot">Bode gain vs. frequency plot</a>. That design has the <a href="http://en.wikipedia.org/wiki/Rule_of_thumb" title="Rule of thumb">rule of thumb</a>
 built-in safety margin to deal with non-ideal realities like multiple 
poles (or zeros), nonlinearity (signal amplitude dependence) and 
manufacturing variations, any of which can lead to too much overshoot. 
The adjustment of the pole separation (that is, setting&nbsp;α) is the 
subject of <a href="http://en.wikipedia.org/wiki/Frequency_compensation" title="Frequency compensation">frequency compensation</a>, and one such method is <a href="http://en.wikipedia.org/wiki/Pole_splitting" title="Pole splitting">pole splitting</a>.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Step_response&amp;action=edit&amp;section=6" title="Edit section: Control of settling time">edit</a>]</span> <span class="mw-headline" id="Control_of_settling_time">Control of settling time</span></h3>
<p>The amplitude of ringing in the step response in Figure 3 is governed
 by the damping factor exp ( −ρ t ). That is, if we specify some 
acceptable step response deviation from final value, say Δ, that is:</p>
<dl>
<dd><img class="tex" alt=" S(t)  \le 1 + \Delta, \, " src="wikipedia-Step_response_pliki/ac757f5ff84a55d2046bb5f9e10ff187.png"></dd>
</dl>
<p>this condition is satisfied regardless of the value of β <i>A</i><sub>OL</sub> provided the time is longer than the settling time, say <i>t</i><sub>S</sub>, given by:<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span>[</span>3<span>]</span></a></sup></p>
<dl>
<dd><img class="tex" alt=" \Delta = e^{- \rho t_S }\text{ or }t_S = \frac { \ln \left( \frac{1} { \Delta} \right) } { \rho } = \tau_2 \frac {2 \ln \left( \frac{1} { \Delta} \right) } { 1 + \frac { \tau_2 } { \tau_1} } \approx 2 \tau_2 \ln \left( \frac{1} { \Delta} \right), " src="wikipedia-Step_response_pliki/6aa355a1435560d9009de007728cec66.png"></dd>
</dl>
<p>where the approximation τ<sub>1</sub>&nbsp;&gt;&gt;&nbsp;τ<sub>2</sub> is applicable because of the overshoot control condition, which makes τ<sub>1</sub>&nbsp;=&nbsp;<i>αβA</i><sub>OL</sub> τ<sub>2</sub>.
 Often the settling time condition is referred to by saying the settling
 period is inversely proportional to the unity gain bandwidth, because 
1/(2π&nbsp;τ<sub>2</sub>) is close to this bandwidth for an amplifier with typical <a href="http://en.wikipedia.org/wiki/Frequency_compensation#Dominant-pole_compensation" title="Frequency compensation">dominant pole compensation</a>. However, this result is more precise than this <a href="http://en.wikipedia.org/wiki/Rule_of_thumb" title="Rule of thumb">rule of thumb</a>. As an example of this formula, if Δ&nbsp;=&nbsp;1/e<sup>4</sup>&nbsp;=&nbsp;1.8&nbsp;%, the settling time condition is <i>t</i><sub>S</sub>&nbsp;=&nbsp;8&nbsp;τ<sub>2</sub>.</p>
<p>In general, control of overshoot sets the time constant ratio, and settling time <i>t</i><sub>S</sub> sets&nbsp;τ<sub>2</sub>.<sup id="cite_ref-Johns_3-0" class="reference"><a href="#cite_note-Johns-3"><span>[</span>4<span>]</span></a></sup><sup id="cite_ref-Sansen_4-0" class="reference"><a href="#cite_note-Sansen-4"><span>[</span>5<span>]</span></a></sup> <sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span>[</span>6<span>]</span></a></sup></p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Step_response&amp;action=edit&amp;section=7" title="Edit section: Phase margin">edit</a>]</span> <span class="mw-headline" id="Phase_margin">Phase margin</span></h3>
<div class="thumb tright">
<div class="thumbinner" style="width:282px;"><a href="http://en.wikipedia.org/wiki/File:Phase_for_Step_Response.PNG" class="image"><img alt="" src="wikipedia-Step_response_pliki/280px-Phase_for_Step_Response.PNG" class="thumbimage" width="280" height="244"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Phase_for_Step_Response.PNG" class="internal" title="Enlarge"><img src="wikipedia-Step_response_pliki/magnify-clip.png" alt="" width="15" height="11"></a></div>
Figure 5: Bode gain plot to find phase margin; scales are logarithmic, 
so labeled separations are multiplicative factors. For example, <i>f</i><sub>0&nbsp;dB</sub>&nbsp;=&nbsp;<i>βA</i><sub>0</sub>&nbsp;×&nbsp;<i>f</i><sub>1</sub>.</div>
</div>
</div>
<p>Next, the choice of pole ratio τ<sub>1</sub>/τ<sub>2</sub> is related to the phase margin of the feedback amplifier.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span>[</span>7<span>]</span></a></sup> The procedure outlined in the <a href="http://en.wikipedia.org/wiki/Bode_plot#Examples_using_Bode_plots" title="Bode plot">Bode plot</a>
 article is followed. Figure 5 is the Bode gain plot for the two-pole 
amplifier in the range of frequencies up to the second pole position. 
The assumption behind Figure 5 is that the frequency <i>f</i><sub>0&nbsp;dB</sub> lies between the lowest pole at <i>f</i><sub>1</sub>&nbsp;=&nbsp;1/(2πτ<sub>1</sub>) and the second pole at <i>f</i><sub>2</sub>&nbsp;=&nbsp;1/(2πτ<sub>2</sub>). As indicated in Figure 5, this condition is satisfied for values of&nbsp;α&nbsp;≥&nbsp;1.</p>
<p>Using Figure 5 the frequency (denoted by <i>f</i><sub>0&nbsp;dB</sub>) is found where the loop gain β<i>A</i><sub>0</sub> satisfies the unity gain or 0&nbsp;dB condition, as defined by:</p>
<dl>
<dd><img class="tex" alt=" | \beta A_\text{OL} ( f_\text{0 db} ) | = 1. \ " src="wikipedia-Step_response_pliki/a0373767c97e0ee8293923b4e84f7685.png"></dd>
</dl>
<p>The slope of the downward leg of the gain plot is (20 dB/decade); for
 every factor of ten increase in frequency, the gain drops by the same 
factor:<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span>[</span>8<span>]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span>[</span>9<span>]</span></a></sup></p>
<dl>
<dd><img class="tex" alt=" f_\text{0 dB} = \beta A_0 f_1. \, " src="wikipedia-Step_response_pliki/0874f1ca4f6a24969a16ad5764c1f34c.png"></dd>
</dl>
<p>The phase margin is the departure of the phase at <i>f</i><sub>0&nbsp;dB</sub> from −180°. Thus, the margin is:</p>
<dl>
<dd><img class="tex" alt=" \phi_m = 180 ^\circ - \arctan (f_\text{0 dB} /f_1) - \arctan ( f_\text{0 dB} /f_2). \ " src="wikipedia-Step_response_pliki/6c49968d426dc9c3bebc19924c6ec096.png"></dd>
</dl>
<p>Because <i>f</i><sub>0&nbsp;dB</sub> / <i>f</i><sub>1</sub> =&nbsp;<i>βA</i><sub>0</sub>&nbsp;&gt;&gt;&nbsp;1, the term in <i>f</i><sub>1</sub> is 90°. That makes the phase margin:</p>
<dl>
<dd><img class="tex" alt=" \phi_m = 90 ^\circ - \arctan ( f_\text{0 dB} /f_2) \, " src="wikipedia-Step_response_pliki/c44e5da94d3f75a37c75a07c03a7859e.png">
<dl>
<dd><img class="tex" alt=" = 90 ^\circ -  \arctan \left( \frac {\beta A_0 f_1} {\alpha \beta A_0 f_1 } \right) " src="wikipedia-Step_response_pliki/784cce6efe31dca2390760195cac9f0f.png"></dd>
<dd><img class="tex" alt=" = 90 ^\circ - \arctan \left( \frac {1} {\alpha } \right) = \arctan \left(  \alpha  \right). " src="wikipedia-Step_response_pliki/5a2b612a16b09e8ef305bacd60d4b573.png"></dd>
</dl>
</dd>
</dl>
<p>In particular, for case α = 1, φ<sub>m</sub> = 45°, and for α = 2, φ<sub>m</sub> = 63.4°. Sansen<sup id="cite_ref-Sansen3_9-0" class="reference"><a href="#cite_note-Sansen3-9"><span>[</span>10<span>]</span></a></sup> recommends α = 3, φ<sub>m</sub> = 71.6° as a "good safety position to start with".</p>
<p>If α is increased by shortening τ<sub>2</sub>, the settling time <i>t</i><sub>S</sub> also is shortened. If α is increased by lengthening τ<sub>1</sub>, the settling time <i>t</i><sub>S</sub> is little altered. More commonly, both τ<sub>1</sub> <i>and</i> τ<sub>2</sub> change, for example if the technique of <a href="http://en.wikipedia.org/wiki/Pole_splitting" title="Pole splitting">pole splitting</a> is used.</p>
<p>As an aside, for an amplifier with more than two poles, the diagram 
of Figure 5 still may be made to fit the Bode plots by making <i>f</i><sub>2</sub> a fitting parameter, referred to as an "equivalent second pole" position.<sup id="cite_ref-Palumbo_10-0" class="reference"><a href="#cite_note-Palumbo-10"><span>[</span>11<span>]</span></a></sup></p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Step_response&amp;action=edit&amp;section=8" title="Edit section: Formal mathematical description">edit</a>]</span> <span class="mw-headline" id="Formal_mathematical_description">Formal mathematical description</span></h2>
<div class="thumb tright">
<div class="thumbinner" style="width:342px;"><a href="http://en.wikipedia.org/wiki/File:Step_response.jpg" class="image"><img alt="" src="wikipedia-Step_response_pliki/Step_response.jpg" class="thumbimage" width="340" height="100"></a>
<div class="thumbcaption">Figure 4: Black box representation of a dynamical system, its input and its step response.</div>
</div>
</div>
<p>This section provides a formal mathematical definition of step response in terms of the abstract mathematical concept of a <a href="http://en.wikipedia.org/wiki/Dynamical_system_%28definition%29" title="Dynamical system (definition)">dynamical system</a> <img class="tex" alt="\scriptstyle\mathfrak{S}" src="wikipedia-Step_response_pliki/fc11ab0a24e7bdf5acbd71d7135f38d6.png">: all notations and assumptions required for the following description are listed here.</p>
<ul>
<li><img class="tex" alt="\scriptstyle t\in T" src="wikipedia-Step_response_pliki/62c2ecd325d3269d7b9f0501cd64af49.png"> is the <a href="http://en.wikipedia.org/wiki/Dynamical_system_%28definition%29" title="Dynamical system (definition)">evolution parameter</a> of the system, called "<a href="http://en.wikipedia.org/wiki/Time" title="Time">time</a>" for the sake of simplicity,</li>
<li><img class="tex" alt="\scriptstyle\boldsymbol{x}|_t\in M" src="wikipedia-Step_response_pliki/11936687c5b5b1cc4526503f736bfff2.png"> is the <a href="http://en.wikipedia.org/wiki/Dynamical_system_%28definition%29" title="Dynamical system (definition)">state</a> of the system at time <img class="tex" alt="t\," src="wikipedia-Step_response_pliki/0c68620ee2ea4f1286fcd672a47ea080.png">, called "output" for the sake of simplicity,</li>
<li><img class="tex" alt="\scriptstyle\Phi:T\times M\longrightarrow M" src="wikipedia-Step_response_pliki/ef25fc22897979871175f68d0f966caf.png"> is the dynamical system <a href="http://en.wikipedia.org/wiki/Dynamical_system_%28definition%29" title="Dynamical system (definition)">evolution function</a>,</li>
<li><img class="tex" alt="\scriptstyle\Phi(0,\boldsymbol{x})=\boldsymbol{x}_0\in M" src="wikipedia-Step_response_pliki/a0cf289a66c3956a024952eb85af9133.png"> is the dynamical system <a href="http://en.wikipedia.org/wiki/Dynamical_system_%28definition%29" title="Dynamical system (definition)">initial state</a>,</li>
<li><img class="tex" alt="\scriptstyle H(t)\," src="wikipedia-Step_response_pliki/0e51a5550991c4bf38cefe5a85c6f9a2.png"> is the <a href="http://en.wikipedia.org/wiki/Heaviside_step_function" title="Heaviside step function">Heaviside step function</a></li>
</ul>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Step_response&amp;action=edit&amp;section=9" title="Edit section: Nonlinear dynamical system">edit</a>]</span> <span class="mw-headline" id="Nonlinear_dynamical_system">Nonlinear dynamical system</span></h3>
<p>For a general dynamical system, the step response is defined as follows:</p>
<dl>
<dd>
<dl>
<dd><img class="tex" alt=" \boldsymbol{x}|_t = \Phi_{\{H(t)\} \left(t,{\boldsymbol{x}_0} \right)}. \, " src="wikipedia-Step_response_pliki/6dd3f73ec0f52fa08e7d586044f9bdd2.png"></dd>
</dl>
</dd>
</dl>
<p>It is the <a href="http://en.wikipedia.org/wiki/Dynamical_system_%28definition%29" title="Dynamical system (definition)">evolution function</a> when the control inputs (or <a href="http://en.wikipedia.org/wiki/Linear_differential_equation" title="Linear differential equation">source term</a>, or <a href="http://en.wikipedia.org/w/index.php?title=Forcing_input&amp;action=edit&amp;redlink=1" class="new" title="Forcing input (page does not exist)">forcing inputs</a>) are Heaviside functions: the notation emphasizes this concept showing <i>H</i>(<i>t</i>) as a subscript.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Step_response&amp;action=edit&amp;section=10" title="Edit section: Linear dynamical system">edit</a>]</span> <span class="mw-headline" id="Linear_dynamical_system">Linear dynamical system</span></h3>
<p>For a <a href="http://en.wikipedia.org/wiki/Linear_system" title="Linear system">linear</a> <a href="http://en.wikipedia.org/wiki/Time-invariant_system" title="Time-invariant system">time-invariant</a> black box, let <img class="tex" alt="\scriptstyle\mathfrak{{S}}\ \equiv \ S" src="wikipedia-Step_response_pliki/b72726a70d0e95a51b282f164e47156f.png"> for notational convenience: the step response can be obtained by <a href="http://en.wikipedia.org/wiki/Convolution" title="Convolution">convolution</a> of the <a href="http://en.wikipedia.org/wiki/Heaviside_step_function" title="Heaviside step function">Heaviside step function</a> control and the <a href="http://en.wikipedia.org/wiki/Impulse_response" title="Impulse response">impulse response</a> <i>h</i>(<i>t</i>) of the system itself</p>
<dl>
<dd>
<dl>
<dd><img class="tex" alt="a(t) = {h*H}(t) = {H*h}(t) = \int_{-\infty }^{+\infty} h(\tau) H(t - \tau ) \, d\tau = \int_{-\infty}^t h(\tau) \, d\tau." src="wikipedia-Step_response_pliki/69578302339f1b9f0ca01ff13b5396dc.png"></dd>
</dl>
</dd>
</dl>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Step_response&amp;action=edit&amp;section=11" title="Edit section: See also">edit</a>]</span> <span class="mw-headline" id="See_also">See also</span></h2>
<ul>
<li><a href="http://en.wikipedia.org/wiki/Impulse_response" title="Impulse response">Impulse response</a></li>
<li><a href="http://en.wikipedia.org/wiki/Overshoot_%28signal%29" title="Overshoot (signal)">Overshoot (signal)</a></li>
<li><a href="http://en.wikipedia.org/wiki/Pole_splitting" title="Pole splitting">Pole splitting</a></li>
<li><a href="http://en.wikipedia.org/wiki/Rise_time" title="Rise time">Rise time</a></li>
<li><a href="http://en.wikipedia.org/wiki/Settling_time" title="Settling time">Settling time</a></li>
<li><a href="http://en.wikipedia.org/wiki/Time_constant" title="Time constant">Time constant</a></li>
</ul>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Step_response&amp;action=edit&amp;section=12" title="Edit section: References and notes">edit</a>]</span> <span class="mw-headline" id="References_and_notes">References and notes</span></h2>
<div class="reflist references-column-count references-column-count-2" style="-moz-column-count: 2; -webkit-column-count: 2; column-count: 2; list-style-type: decimal;">
<ol class="references">
<li id="cite_note-Kuo-0"><b><a href="#cite_ref-Kuo_0-0">^</a></b> <span class="reference-text"><span class="citation book">Benjamin C Kuo &amp; Golnaraghi F (2003). <a rel="nofollow" class="external text" href="http://worldcat.org/isbn/0-471-13476-7"><i>Automatic control systems</i></a> (Eighth Edition ed.). New York: Wiley. pp.&nbsp;253. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-471-13476-7" title="Special:BookSources/0-471-13476-7">0-471-13476-7</a><span class="printonly">. <a rel="nofollow" class="external free" href="http://worldcat.org/isbn/0-471-13476-7">http://worldcat.org/isbn/0-471-13476-7</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Automatic+control+systems&amp;rft.aulast=Benjamin+C+Kuo+%26+Golnaraghi+F&amp;rft.au=Benjamin+C+Kuo+%26+Golnaraghi+F&amp;rft.date=2003&amp;rft.pages=pp.%26nbsp%3B253&amp;rft.edition=Eighth+Edition&amp;rft.place=New+York&amp;rft.pub=Wiley&amp;rft.isbn=0-471-13476-7&amp;rft_id=http%3A%2F%2Fworldcat.org%2Fisbn%2F0-471-13476-7&amp;rfr_id=info:sid/en.wikipedia.org:Step_response"><span style="display: none;">&nbsp;</span></span></span></li>
<li id="cite_note-Kuo2-1"><b><a href="#cite_ref-Kuo2_1-0">^</a></b> <span class="reference-text"><span class="citation book">Benjamin C Kuo &amp; Golnaraghi F. <a rel="nofollow" class="external text" href="http://worldcat.org/isbn/0-471-13476-7"><i>p. 259</i></a>. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-471-13476-7" title="Special:BookSources/0-471-13476-7">0-471-13476-7</a><span class="printonly">. <a rel="nofollow" class="external free" href="http://worldcat.org/isbn/0-471-13476-7">http://worldcat.org/isbn/0-471-13476-7</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=p.+259&amp;rft.aulast=Benjamin+C+Kuo+%26+Golnaraghi+F&amp;rft.au=Benjamin+C+Kuo+%26+Golnaraghi+F&amp;rft.isbn=0-471-13476-7&amp;rft_id=http%3A%2F%2Fworldcat.org%2Fisbn%2F0-471-13476-7&amp;rfr_id=info:sid/en.wikipedia.org:Step_response"><span style="display: none;">&nbsp;</span></span></span></li>
<li id="cite_note-2"><b><a href="#cite_ref-2">^</a></b> <span class="reference-text">This estimate is a bit conservative (long) because the factor 1 /sin(φ) in the overshoot contribution to <i>S</i> ( <i>t</i> ) has been replaced by 1 /sin(φ) ≈ 1.</span></li>
<li id="cite_note-Johns-3"><b><a href="#cite_ref-Johns_3-0">^</a></b> <span class="reference-text"><span class="citation book">David A. Johns &amp; Martin K W (1997). <a rel="nofollow" class="external text" href="http://worldcat.org/isbn/0-471-14448-7"><i>Analog integrated circuit design</i></a>. New York: Wiley. pp.&nbsp;234–235. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-471-14448-7" title="Special:BookSources/0-471-14448-7">0-471-14448-7</a><span class="printonly">. <a rel="nofollow" class="external free" href="http://worldcat.org/isbn/0-471-14448-7">http://worldcat.org/isbn/0-471-14448-7</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Analog+integrated+circuit+design&amp;rft.aulast=David+A.+Johns+%26+Martin+K+W&amp;rft.au=David+A.+Johns+%26+Martin+K+W&amp;rft.date=1997&amp;rft.pages=pp.%26nbsp%3B234%E2%80%93235&amp;rft.place=New+York&amp;rft.pub=Wiley&amp;rft.isbn=0-471-14448-7&amp;rft_id=http%3A%2F%2Fworldcat.org%2Fisbn%2F0-471-14448-7&amp;rfr_id=info:sid/en.wikipedia.org:Step_response"><span style="display: none;">&nbsp;</span></span></span></li>
<li id="cite_note-Sansen-4"><b><a href="#cite_ref-Sansen_4-0">^</a></b> <span class="reference-text"><span class="citation book">Willy M C Sansen (2006). <a rel="nofollow" class="external text" href="http://worldcat.org/isbn/0-387-25746-2"><i>Analog design essentials</i></a>. Dordrecht, The Netherlands: Springer. p.&nbsp;§0528 p. 163. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-387-25746-2" title="Special:BookSources/0-387-25746-2">0-387-25746-2</a><span class="printonly">. <a rel="nofollow" class="external free" href="http://worldcat.org/isbn/0-387-25746-2">http://worldcat.org/isbn/0-387-25746-2</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Analog+design+essentials&amp;rft.aulast=Willy+M+C+Sansen&amp;rft.au=Willy+M+C+Sansen&amp;rft.date=2006&amp;rft.pages=p.%26nbsp%3B%C2%A70528+p.+163&amp;rft.place=Dordrecht%2C+The+Netherlands&amp;rft.pub=Springer&amp;rft.isbn=0-387-25746-2&amp;rft_id=http%3A%2F%2Fworldcat.org%2Fisbn%2F0-387-25746-2&amp;rfr_id=info:sid/en.wikipedia.org:Step_response"><span style="display: none;">&nbsp;</span></span></span></li>
<li id="cite_note-5"><b><a href="#cite_ref-5">^</a></b> <span class="reference-text">According to Johns and Martin, <i>op. cit.</i>, settling time is significant in <a href="http://en.wikipedia.org/wiki/Switched_capacitor" title="Switched capacitor">switched-capacitor circuits</a>, for example, where an op amp settling time must be less than half a clock period for sufficiently rapid charge transfer.</span></li>
<li id="cite_note-6"><b><a href="#cite_ref-6">^</a></b> <span class="reference-text">The
 gain margin of the amplifier cannot be found using a two-pole model, 
because gain margin requires determination of the frequency <i>f</i><sub>180</sub> where the gain flips sign, and this never happens in a two-pole system. If we know <i>f</i><sub>180</sub> for the amplifier at hand, the gain margin can be found approximately, but <i>f</i><sub>180</sub>
 then depends on the third and higher pole positions, as does the gain 
margin, unlike the estimate of phase margin, which is a two-pole 
estimate.</span></li>
<li id="cite_note-7"><b><a href="#cite_ref-7">^</a></b> <span class="reference-text">For more detail on the use of logarithmic scales, see <a href="http://en.wikipedia.org/wiki/Logarithmic_scale#Slope_of_a_log-log_plot" title="Logarithmic scale">log scale</a>.</span></li>
<li id="cite_note-8"><b><a href="#cite_ref-8">^</a></b> <span class="reference-text">See also <a href="http://en.wikipedia.org/wiki/Pole_splitting#Selection_of_CC" title="Pole splitting">Figure 3 in pole splitting</a>.</span></li>
<li id="cite_note-Sansen3-9"><b><a href="#cite_ref-Sansen3_9-0">^</a></b> <span class="reference-text"><span class="citation book">Willy M C Sansen. <a rel="nofollow" class="external text" href="http://worldcat.org/isbn/0-387-25746-2"><i>§0526 p. 162</i></a>. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-387-25746-2" title="Special:BookSources/0-387-25746-2">0-387-25746-2</a><span class="printonly">. <a rel="nofollow" class="external free" href="http://worldcat.org/isbn/0-387-25746-2">http://worldcat.org/isbn/0-387-25746-2</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=%C2%A70526+p.+162&amp;rft.aulast=Willy+M+C+Sansen&amp;rft.au=Willy+M+C+Sansen&amp;rft.isbn=0-387-25746-2&amp;rft_id=http%3A%2F%2Fworldcat.org%2Fisbn%2F0-387-25746-2&amp;rfr_id=info:sid/en.wikipedia.org:Step_response"><span style="display: none;">&nbsp;</span></span></span></li>
<li id="cite_note-Palumbo-10"><b><a href="#cite_ref-Palumbo_10-0">^</a></b> <span class="reference-text"><span class="citation book">Gaetano Palumbo &amp; Pennisi S (2002). <a rel="nofollow" class="external text" href="http://books.google.com/books?id=Xb0W1VsQFe0C&amp;pg=PA98&amp;dq=%22equivalent+two-pole+amplifier%22"><i>Feedback amplifiers: theory and design</i></a>. Boston/Dordrecht/London: Kluwer Academic Press. pp.&nbsp;§ 4.4 pp. 97–98. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-7923-7643-9" title="Special:BookSources/0-7923-7643-9">0-7923-7643-9</a><span class="printonly">. <a rel="nofollow" class="external free" href="http://books.google.com/books?id=Xb0W1VsQFe0C&amp;pg=PA98&amp;dq=%22equivalent+two-pole+amplifier%22">http://books.google.com/books?id=Xb0W1VsQFe0C&amp;pg=PA98&amp;dq=%22equivalent+two-pole+amplifier%22</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Feedback+amplifiers%3A+theory+and+design&amp;rft.aulast=Gaetano+Palumbo+%26+Pennisi+S&amp;rft.au=Gaetano+Palumbo+%26+Pennisi+S&amp;rft.date=2002&amp;rft.pages=pp.%26nbsp%3B%C2%A7+4.4+pp.+97%E2%80%9398&amp;rft.place=Boston%2FDordrecht%2FLondon&amp;rft.pub=Kluwer+Academic+Press&amp;rft.isbn=0-7923-7643-9&amp;rft_id=http%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DXb0W1VsQFe0C%26pg%3DPA98%26dq%3D%2522equivalent%2Btwo-pole%2Bamplifier%2522&amp;rfr_id=info:sid/en.wikipedia.org:Step_response"><span style="display: none;">&nbsp;</span></span></span></li>
</ol>
</div>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Step_response&amp;action=edit&amp;section=13" title="Edit section: Further reading">edit</a>]</span> <span class="mw-headline" id="Further_reading">Further reading</span></h2>
<div class="refbegin" style="">
<ul>
<li>Robert I. Demrow <i>Settling time of operational amplifiers</i> <a rel="nofollow" class="external autonumber" href="http://www.analog.com/static/imported-files/application_notes/466359863287538299597392756AN359.pdf">[1]</a></li>
<li>Cezmi Kayabasi <i>Settling time measurement techniques achieving high precision at high speeds</i> <a rel="nofollow" class="external autonumber" href="http://www.wpi.edu/Pubs/ETD/Available/etd-050505-140358/unrestricted/ckayabasi.pdf">[2]</a></li>
<li>Vladimir Igorevic Arnol'd "Ordinary differential equations", various
 editions from MIT Press and from Springer Verlag, chapter 1 
"Fundamental concepts"</li>
</ul>
</div>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Step_response&amp;action=edit&amp;section=14" title="Edit section: External links">edit</a>]</span> <span class="mw-headline" id="External_links">External links</span></h2>
<ul>
<li><a rel="nofollow" class="external text" href="http://bcs.wiley.com/he-bcs/Books?action=resource&amp;bcsId=2357&amp;itemId=0471134767&amp;resourceId=5596">Kuo power point slides; Chapter 7 especially</a></li>
</ul>


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